Problem 63 - College Algebra and Calculus: An Applied Approach

Question

Use a symbolic integration utility to evaluate the integral. $$ \int_{0}^{5} x^{4}\left(25-x^{2}\right)^{3 / 2} d x $$

Step-by-Step Solution

Answer

The definite integral of the given function from 0 to 5 is 25.

Step 1: Identify Approach- Substitution

Given function can be altered to a form that allows easier computation via substitution. Let us replace \(25 - x^2\) with \(u\), so that the derivative \(-2x\) can be eliminated in the process of substitution.

Step 2: Perform the Substitution

Let's substitute \(u = 25 - x^2\), which gives \(du = -2x \, dx\). Rearranging for \(dx\) gives \(dx = -du/(2x)\). The new limits for \(u\) are \(25 - 0^2 = 25\) (lower limit) and \(25 - 5^2 = 0\) (upper limit). Substitute these values into the integral, which gives \(-\frac{1}{2} \int_{25}^{0} u^{3/2} du\).

Step 3: Evaluate the Integral

This integral can be evaluated using the power rule, which states that the integral of \(u^n\) is \(u^{n+1} /(n+1)\). Applying this gives \(-\frac{1}{2} [(2/5) u^{5/2}]_{25}^{0}\).

Step 4: Substitute u Back

We have to substitute back the u-values to x: \(-\frac{1}{5} [(25 - x^2)^{5/2} ]_{0}^{5}\).

Step 5: Evaluate Definite Integral

The last step is to substitute the limits of the integral into the integral. This gives \(-\frac{1}{5} [(0) - (25^{5/2})] = \frac{125}{5} = 25\).

Key Concepts

Substitution Method in Integration

The substitution method is a powerful technique in calculus, particularly useful when dealing with integrals of more complex functions. It works much like the reverse of the chain rule for derivatives. The idea is to transform the complicated function into a simpler one, which we can integrate easily, by substituting a part of the original function with a new variable.

Choosing the Substitution

In our example, noticing that the function inside the integral, \(25 - x^2\), raised to a power complicates the integral. By introducing the substitution \(u = 25 - x^2\), we simplify the integrand to a form where the integral becomes straightforward to compute.

Dealing with the Differentiation Variable

A crucial step in substitution is to express \(dx\) in terms of \(du\). We calculate the derivative of \(u\) with respect to \(x\) to find \(du\), and then solve for \(dx\). This allows us to rewrite the entire integral in terms of \(u\) and \(du\).

Adjusting the Limits

If we're performing a definite integral, we also need to change the limits of integration to reflect our new variable \(u\). This avoids having to substitute back to \(x\) after finding the indefinite integral and simplifies the solution process.

Definite Integral and Its Evaluation

A definite integral has both a start and an end point, called the limits of integration, and it represents the area under the curve of a function on a specified interval. To solve a definite integral, we find the indefinite integral first, which is the antiderivative of the function, and then apply the limits.

Application of the Substitution

After substitution and changing the limits accordingly, we have a new integral \(\int_{25}^{0} u^{3/2} du\) with limits that correspond to the new variable \(u\). Even though the limits seem reversed, which would normally mean the integral would evaluate to a negative area, we note that there's also a negative sign from the substitution. These two negatives will cancel each other out in the final evaluation.

Final Evaluation

The final step in evaluating a definite integral is to plug in the upper limit into the antiderivative, subtract the value obtained by plugging in the lower limit, and interpret the result. In this case, when we substituted back to \(x\) and applied the limits, we found the area represented by the original integral to be 25.

Power Rule for Integration

The power rule for integration is a basic rule that helps us integrate functions of the form \(x^n\). It states that the integral of \(x^n\) with respect to \(x\) is \(\frac{1}{n+1} x^{n+1}\), plus a constant of integration when dealing with an indefinite integral.

Applying the Power Rule

In the given problem, after substituting \(u\) for \(25 - x^2\), we need to integrate \(u\) raised to the power of \(3/2\). By applying the power rule, we get \(u^{5/2}\) divided by \(\frac{5}{2}\), or multiplied by \(\frac{2}{5}\), as the new expression for the antiderivative.

Combining with Other Steps

The use of the power rule simplifies the integrand to a single term, which we then evaluate using the adjusted definite integral limits. The ease of using the power rule makes the substitution method even more effective, allowing us to tackle complex integrals that appear unsolvable at first glance.